Transcendental number

In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a nonzero polynomialequation with integer (or, equivalently, rational) coefficients. The best-known transcendental numbers are π and e. Though only a few classes of transcendental numbers are known (in part because it can be extremely difficult to show that a given number is transcendental), transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the algebraic numbers are countable while the sets of real and complex numbers are both uncountable. All real transcendental numbers are irrational, since all rational numbers are algebraic. The converse is not true: not all irrational numbers are transcendental; e.g., the square root of 2 is irrational but not a transcendental number, since it is a solution of the polynomial equation x2 − 2 = 0. Another irrational number that is not transcendental is the golden ratio, φ{\displaystyle \varphi } or ϕ{\displaystyle \phi }, since it is a solution of the polynomial equation x2 − x − 1 = 0.

History

The name "transcendental" comes from the Latin transcendĕre 'to climb over or beyond, surmount',[1] and was first used for the mathematical concept in Leibniz's 1682 paper in which he proved that sin(x) is not an algebraic function of x.[2][3]Euler, in the 18th century, was probably the first person to define transcendental numbers in the modern sense.[4]

Johann Heinrich Lambert conjectured that e and π were both transcendental numbers in his 1768 paper proving the number π is irrational, and proposed a tentative sketch of a proof of π's transcendence.[5]

in which the nth digit after the decimal point is 1 if n is equal to k! (kfactorial) for some k and 0 otherwise.[7] In other words, the nth digit of this number is 1 only if n is one of the numbers 1! = 1, 2! = 2, 3! = 6, 4! = 24, etc. Liouville showed that this number is what we now call a Liouville number; this essentially means that it can be more closely approximated by rational numbers than can any irrational algebraic number. Liouville showed that all Liouville numbers are transcendental.[8]

The first number to be proven transcendental without having been specifically constructed for the purpose was e, by Charles Hermite in 1873.

In 1874, Georg Cantor proved that the algebraic numbers are countable and the real numbers are uncountable. He also gave a new method for constructing transcendental numbers.[9][10] In 1878, Cantor published a construction that proves there are as many transcendental numbers as there are real numbers.[11] Cantor's work established the ubiquity of transcendental numbers.

In 1900, David Hilbert posed an influential question about transcendental numbers, Hilbert's seventh problem: If a is an algebraic number, that is not zero or one, and b is an irrational algebraic number, is ab necessarily transcendental? The affirmative answer was provided in 1934 by the Gelfond–Schneider theorem. This work was extended by Alan Baker in the 1960s in his work on lower bounds for linear forms in any number of logarithms (of algebraic numbers).[12]

Properties

The set of transcendental numbers is uncountably infinite. Since the polynomials with rational coefficients are countable, and since each such polynomial has a finite number of zeroes, the algebraic numbers must also be countable. However, Cantor's diagonal argument proves that the real numbers (and therefore also the complex numbers) are uncountable. Since the real numbers are the union of algebraic and transcendental numbers, they cannot both be countable. This makes the transcendental numbers uncountable.

Any non-constant algebraic function of a single variable yields a transcendental value when applied to a transcendental argument. For example, from knowing that π is transcendental, it can be immediately deduced that numbers such as 5π, (π − 3)/√2, (√π − √3)8 and (π5 + 7)1/7 are transcendental as well.

However, an algebraic function of several variables may yield an algebraic number when applied to transcendental numbers if these numbers are not algebraically independent. For example, π and (1 − π) are both transcendental, but π + (1 − π) = 1 is obviously not. It is unknown whether π + e, for example, is transcendental, though at least one of π + e and πe must be transcendental. More generally, for any two transcendental numbers a and b, at least one of a + b and ab must be transcendental. To see this, consider the polynomial (x − a)(x − b) = x2 − (a + b)x + ab. If (a + b) and ab were both algebraic, then this would be a polynomial with algebraic coefficients. Because algebraic numbers form an algebraically closed field, this would imply that the roots of the polynomial, a and b, must be algebraic. But this is a contradiction, and thus it must be the case that at least one of the coefficients is transcendental.

All Liouville numbers are transcendental, but not vice versa. Any Liouville number must have unbounded partial quotients in its continued fraction expansion. Using a counting argument one can show that there exist transcendental numbers which have bounded partial quotients and hence are not Liouville numbers.

Using the explicit continued fraction expansion of e, one can show that e is not a Liouville number (although the partial quotients in its continued fraction expansion are unbounded). Kurt Mahler showed in 1953 that π is also not a Liouville number. It is conjectured that all infinite continued fractions with bounded terms that are not eventually periodic are transcendental (eventually periodic continued fractions correspond to quadratic irrationals).[13]

3.300330000000000330033... and its reciprocal 0.30300000303..., two numbers with only two different decimal digits whose nonzero digit positions are given by the Moser–de Bruijn sequence and its double.[28]

Numbers which have yet to be proven to be either transcendental or algebraic:

Most sums, products, powers, etc. of the number π and the number e, e.g. π + e, π − e, πe, π/e, ππ, ee, πe, π√2, eπ2 are not known to be rational, algebraic, irrational or transcendental. A notable exception is eπ√n (for any positive integer n) which has been proven transcendental.[29]

So when dividing each integral in P by k!, the initial one is not divisible by k+1, but all the others are, as long as k+1 is prime and larger than n and |c0|. It follows that Pk!{\displaystyle {\tfrac {P}{k!}}} itself is not divisible by the prime k+1 and therefore cannot be zero.

where u(x){\displaystyle u(x)} and v(x){\displaystyle v(x)} are continuous functions of x{\displaystyle x} for all x{\displaystyle x}, so are bounded on the interval [0,n]{\displaystyle [0,n]}. That is, there are constants G,H>0{\displaystyle G,H>0} such that

Choosing a value of k{\displaystyle k} satisfying both lemmas leads to a non-zero integer (P/k!{\displaystyle P/k!}) added to a vanishingly small quantity (Q/k!{\displaystyle Q/k!}) being equal to zero, is an impossibility. It follows that the original assumption, that e{\displaystyle e} can satisfy a polynomial equation with integer coefficients, is also impossible; that is, e{\displaystyle e} is transcendental.

For detailed information concerning the proofs of the transcendence of π and e, see the references and external links.

Mahler's classification

Kurt Mahler in 1932 partitioned the transcendental numbers into 3 classes, called S, T, and U.[30] Definition of these classes draws on an extension of the idea of a Liouville number (cited above).

Measure of irrationality of a real number

One way to define a Liouville number is to consider how small a given real number x makes linear polynomials |qx − p| without making them exactly 0. Here p, q are integers with |p|, |q| bounded by a positive integer H.

Let m(x, 1, H) be the minimum non-zero absolute value these polynomials take and take:

ω(x, 1) is often called the measure of irrationality of a real number x. For rational numbers, ω(x, 1) = 0 and is at least 1 for irrational real numbers. A Liouville number is defined to have infinite measure of irrationality. Roth's theorem says that irrational real algebraic numbers have measure of irrationality 1.

Measure of transcendence of a complex number

Next consider the values of polynomials at a complex number x, when these polynomials have integer coefficients, degree at most n, and height at most H, with n, H being positive integers.

Let m(x,n,H) be the minimum non-zero absolute value such polynomials take at x and take:

ω(x) is often called the measure of transcendence of x. If the ω(x,n) are bounded, then ω(x) is finite, and x is called an S number. If the ω(x,n) are finite but unbounded, x is called a T number. x is algebraic if and only if ω(x) = 0.

Clearly the Liouville numbers are a subset of the U numbers. William LeVeque in 1953 constructed U numbers of any desired degree.[31][32] The Liouville numbers and hence the U numbers are uncountable sets. They are sets of measure 0.[33]

T numbers also comprise a set of measure 0.[34] It took about 35 years to show their existence. Wolfgang M. Schmidt in 1968 showed that examples exist. However, almost all complex numbers are S numbers.[35] Mahler proved that the exponential function sends all non-zero algebraic numbers to S numbers:[36][37] this shows that e is an S number and gives a proof of the transcendence of π. The most that is known about π is that it is not a U number. Many other transcendental numbers remain unclassified.

Two numbers x, y are called algebraically dependent if there is a non-zero polynomial P in 2 indeterminates with integer coefficients such that P(x, y) = 0. There is a powerful theorem that 2 complex numbers that are algebraically dependent belong to the same Mahler class.[32][38] This allows construction of new transcendental numbers, such as the sum of a Liouville number with e or π.

It is often speculated that S stood for the name of Mahler's teacher Carl Ludwig Siegel and that T and U are just the next two letters.

Koksma's equivalent classification

Jurjen Koksma in 1939 proposed another classification based on approximation by algebraic numbers.[30][39]

Consider the approximation of a complex number x by algebraic numbers of degree ≤ n and height ≤ H. Let α be an algebraic number of this finite set such that |x − α| has the minimum positive value. Define ω*(x,H,n) and ω*(x,n) by:

LeVeque's construction

It can be shown that the nth root of λ (a Liouville number) is a U-number of degree n.[40]

This construction can be improved to create an uncountable family of U-numbers of degree n. Let Z be the set consisting of every other power of 10 in the series above for λ. The set of all subsets of Z is uncountable. Deleting any of the subsets of Z from the series for λ creates uncountably many distinct Liouville numbers, whose nth roots are U-numbers of degree n.

Type

The supremum of the sequence {ω(x, n)} is called the type. Almost all real numbers are S numbers of type 1, which is minimal for real S numbers. Almost all complex numbers are S numbers of type 1/2, which is also minimal. The claims of almost all numbers were conjectured by Mahler and in 1965 proved by Vladimir Sprindzhuk.[31]

^Allouche & Shallit (2003) pp.385,403; The name 'Fredholm number' is misplaced: Kempner first proved this number is transcendental, and the note on page 403 states that Fredholm never studied this number